Effect of load inclination on the undrained bearing capacity of surface spread footings above voids

Computers and Geotechnics 66 (2015) 245–252

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Technical Communication

Effect of load inclination on the undrained bearing capacity of surface spread footings above voids Joon Kyu Lee, Sangseom Jeong ⇑, Junyoung Ko Dept. of Civil and Environmental Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, Republic of Korea

a r t i c l e

i n f o

Article history: Received 13 October 2014 Received in revised form 4 February 2015 Accepted 5 February 2015 Available online 20 February 2015 Keywords: Bearing capacity Void Inclined loading Spread footing Finite element analysis Clay

a b s t r a c t Footings are often situated on ground that includes voids that were either undetected or had not formed at the time of construction. In this study, small-strain finite element analyses are conducted to investigate the influence of load inclination on the bearing capacity of surface spread footings on undrained homogeneous clay with single and double continuous voids. The numerical solutions are compared with existing theoretical and empirical predictions. The results are presented as failure envelopes expressed in terms of loads that are non-dimensionalised by the footing width and the undrained soil shear strength (these are known as bearing capacity factors) and loads that are normalized by the uniaxial ultimate load. If underground voids exist under the footing, the shape and the size of the failure envelope defining the undrained bearing capacity of the footing subjected to combined horizontal and vertical loading depends on the location, the geometry and the number of voids. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Underground voids occur as a result of the dissolution of soluble material in response to hydrochemical changes in the soil and/or rock [1]. A karstic cavity in a limestone formation/deposit is a typical example of mineral dissolution. Unstable voids may also form because of human activity such as mining or leakage from a faulty sewer [2]. The presence of underground voids beneath a rigid foundation system has a significant effect on the stability of the foundation, which can lead to extensive damage and loss of life. Several studies have been undertaken to evaluate the stability of voids embedded within cohesive-frictional soil using model tests [3–6], finite element simulations [7–11], or plasticity limit analyses [12,13]. These analyses concentrated on the vertical footing capacity because the vertical load on a footing, which is mainly due to the weight of the superstructure, is the most important. However, for footings under inclined loading, the previous studies may not provide reliable predictions. The objective of this study is to explore the effect of load inclination on the bearing capacity of spread footings on the surface of homogeneous, purely cohesive soil with voids. The effects of the location, the shape and the number (i.e., one or two) of continuous voids are studied. Finite element solutions are obtained, and the

⇑ Corresponding author. Tel.: +82 2 2123 2807; fax: +82 2 364 5300. E-mail address: [email protected] (S. Jeong). http://dx.doi.org/10.1016/j.compgeo.2015.02.003 0266-352X/Ó 2015 Elsevier Ltd. All rights reserved.

results are presented as normalized failure envelopes in the horizontal and vertical loading planes. 2. Background For undrained conditions, the bearing capacity of a rigid spread footing at the surface subjected to inclined loading can be approximated as

qu ¼ cu Nc fi

ð1Þ

where qu is the ultimate (average) bearing stress on the footing, cu is the undrained shear strength of the soil, Nc is the bearing capacity factor for cohesion, and fi is the load inclination factor. For pure vertical loading, the load inclination factor has a value of 1 and the solution is equal to the Prandtl solution [14]. Table 1 summarizes the most commonly used expressions for the load inclination factor, which were derived by Hansen [15], Meyerhof [16] and Vesic [17] using limit analysis, a theoretically rigorous method for calculating the bearing capacity of footings. Green [18] suggested a plane-strain plasticity solution for a surface footing under inclined loading given by

   V p H þ cos1 þ ¼ 1þ Bcu Bcu 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 H 1 Bcu

ð2Þ

where B is the width of the footing and V and H are the vertical and horizontal components of the applied load, respectively. For a

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footing on undrained clay, Murff [19] proposed a general form of the three-dimensional failure locus that can be simplified for combined horizontal and vertical loading as

Table 1 Load inclination factors fi for bearing capacity of footing. Hansen [15]

Meyerhof [16]

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi fi ¼ 0:5 þ 0:5 1  BcHu

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   V H ¼ 1 Bcu Bcu

ð3Þ





h fi ¼ 1  90 

2

Vesic [17] fi ¼ 1  Bc2H u Nc

Note: B is the footing width; H is the horizontal load; h is the load inclination angle; cu is the soil undrained shear strength; and Nc is the bearing capacity factor for cohesion.

3. Problem definition A general layout of the problem to be analysed is shown in Fig. 1. A rigid spread footing of width B is situated on an isotropic, homogenous clay with an undrained shear strength cu, a Young’s modulus Eu, and a unit weight c (Fig. 1(a)). The footing is subjected to a centred load with an inclination angle h, which is positive in the clockwise direction. The inclination angle ranges between 90° and +90°. The inclined load can be converted a statically equivalent system consisting of horizontal and vertical components, i.e., H and V. Continuous voids in the clay parallel to the axis of the footing are assumed. The bearing capacity of a footing above the voids depends on the shape, the location, and the number of voids [8,10]. For a single void, the void is assumed to be either square and rectangular in shape, where the geometry is defined by two dimensionless parameters, the void width n and the void height m. A similar assumption was used in Abbo et al. [20]. The location of a void relative to the footing is given by the vertical void distance a and the horizontal void distance b. The radial void distance k is defined as the ratio of the distance between the centres of a reference void and the actual void to the footing width, where the reference void has the distance parameters a = 1.5 and b = 0. The parameters have the following values: 1.5 6 a 6 4.0, 0 6 b 6 3.0, and 0 6 k 6 2.5 in increments of 0.5. For two (i.e., double) voids, the two parallel square voids are separated by a distance

V θ

Undrained clay

cu , E u , γ

referred to as the void spacing s, and their configuration can be divided into two types, parallel and symmetrical, as shown in Fig. 1(b).

4. Finite element model All analyses were performed using the commercial software PLAXIS 2D, version 2012 [21]. A small-strain finite-element (FE) model with plane stain was developed to simulate the rigid spread footing, which was progressively displaced until failure occurred. The footing was modelled with six-node triangular plate elements, and the soil was modelled with fifteen-node triangular elements. The response of the undrained soil was prescribed by an isotropic, linearly elastic-perfectly plastic constitutive law using

9.5B

B

9.5B

15B

H

B Rigid strip footing

αB βB

mB

45

λB

Fig. 2. Typical finite element mesh and boundary conditions.

nB

(a) Single void

αB

αB

sB

sB

C.L.

C.L.

Parallel configuration

Symmetrical configuration

(b) Twin voids Fig. 1. Problem notation (modified from Kiyosumi et al. [13]).

Fig. 3. Load–displacement curves for spread footing centered above single void.

J.K. Lee et al. / Computers and Geotechnics 66 (2015) 245–252

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Fig. 4. Comparison of failure envelops for spread footing on ground without voids.

Fig. 6. Failure envelops for varying a values under H–V loading.

Fig. 5. Effect of soil stiffness on failure envelop for H–V loading.

the Tresca yield criterion with an associated flow rule. The elastic behaviour was characterized by the undrained Young’s modulus Eu (=30 MPa) and Poisson’s ratio t (=0.495). The soil stiffness was homogenous with Young’s modulus to undrained shear strength ratios Eu/cu of 100, 300 and 500. The footing was modelled as a non-porous, linearly elastic material with a thickness of 1 m, and Young’s modulus for concrete Ec was assumed to be 30 GPa. The geostatic stress was established by assuming a bulk unit weight c = 20 kN/m3 with K0 equal to unity. It is noteworthy that the undrained load capacity of a surface footing is not sensitive to the soil unit weight [22].

A FE mesh for the footing-above-void system and the applied displacement boundary conditions are shown in Fig. 2. The mesh extended a distance of 9.5 B beyond the edges of the footing and a depth of 15 B beneath the footing, which was found to be sufficient to prevent boundary effects. At the sides of the mesh, no lateral movement was permitted, and full fixity was imposed at the base of the mesh. The void was generated as part of the initial conditions at the target depth for each analysis. Small elements were introduced in the region adjacent to the footing to improve the accuracy of the results. The total number of elements ranged between 7874 and 8582, depending on the geometrical parameters of the voids. For all of the analyses, the shear strength of the footing-soil interface was assumed to be fully rough, permitting unlimited tensile normal stresses between the footing and the soil. A very thin (2.5 cm) layer with zero tension was modelled beneath the footing to allow an effective gap to form between part of the footing and the soil [11]. Based on the stresses predicted in the elements at the interface of the footing and the soil, the horizontal and vertical loads were computed. In this study, a load control analysis was used to find the ultimate bearing capacities of the footing for horizontal and vertical loads and their combination [23]. The inclined load can be increased in gradual steps using nonlinear analysis. The analysis program stops when the load reaches a certain value and large displacements occur in the model. This limit depends on the failure criterion and the soil properties. As an example, the load–displacement response of a footing centred above a single void is plotted in Fig. 3, where w represents the horizontal and vertical displacements (subscripts x and y, respectively). For all cases, the footing exhibited a clear load limit, which was taken as the failure load.

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J.K. Lee et al. / Computers and Geotechnics 66 (2015) 245–252 [Ő10-3 m] 25.00

(a)

-11.33 -47.67 -84.00 -120.33 -156.67 -193.00 -229.33 -265.67 -302.00 -338.33 -374.67 -411.00

V/Bcu = 2.62, H/Bcu = 0.95 θ = +20ƒ, Eu/cu=300 α=1.5, β=0.5, m=1, n=1

-447.33 -483.67 -520.00

[m] 0.00

(b)

-0.16 -0.33 -0.50 -0.67 -0.84 -1.01 -1.18 -1.35 -1.52 -1.69 -1.86

V/Bcu = 2.03, H/Bcu = 0.74 θ = -20ƒ, Eu/cu=300 α=1.5, β=0.5, m=1, n=1

-2.03 -2.20 -2.36 -2.53

Fig. 7. Failure envelops for varying b values under H–V loading.

Fig. 8. FE displacement contours for footing-void system under positive and negative inclined loading.

5. Validation To confirm that the finite element (FE) predictions for the vertical and horizontal bearing capacities of a surface spread footing on soil without voids (B = 2 m, cu = 100 kPa) were reasonable, the results were compared with those of theoretical methods. For pure horizontal loading, the ultimate bearing capacity predicted by the FE analysis was 1.01 Bcu, which compares well with the exact value of 1.00 Bcu. The bearing capacity of the footing under pure vertical loading was estimated by the FE analysis to be 5.16 Bcu, which is 0.4% higher than that obtained using the Pandtl solution, (2 + p) Bcu [14]. The predicted horizontal and vertical failure loads for various inclination angles are presented in Fig. 4, where the existing solutions are included for comparison. The failure loads define a failure envelope in the H–V plane, where load combinations inside the envelope are regarded as safe and those outside the envelope cause the footing to collapse. As shown in the figure, the results of the numerical analyses are very close to those obtained with the exact solution of Green [18] and slightly higher than those suggested by Hansen [15]. The expression from Meyerhof [16] gives values that are largely consistent with the FE results, except for intermediate values of the inclination angle (i.e., 10° < h < 20°). The failure envelopes proposed by Vesic [17] and Murff [19] yield conservative approximations of the numerical failure envelopes. Murff’s solution significantly underestimates the bearing capacity for spread footings subjected to inclined loading. For the expressions given by Meyerhof [16] and Vesic [17], there is a critical inclination angle above which the horizontal capacity of the footing becomes the determining factor. In other words, when the inclination angle is greater than the critical value, the vertical load has little effect

Fig. 9. Failure envelops for varying k values under H–V loading.

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[ 10-3m] -6.00

[m] 0.40

(a)

-0.01

(b)

-9.47

-0.84

-11.20

-1.25

-12.93

-1.67

-14.67

-2.08

-16.40

-2.49

-18.13

-2.91

-19.87

-3.32

-21.60

-3.73

-23.33

-4.15

-25.07

-4.56

V/Bcu = 1.88, H/Bcu = 0.69 θ = +20ƒ, Eu/cu=300 α=1.5, β=0, m=1, n=1

-7.73

-0.43

-26.80

V/Bcu = 2.75, H/Bcu = 1.00 θ = +20ƒ, Eu/cu=300 α=2.5, β=0, m=1, n=1

-4.97 -5.39

-28.53 -30.27

-5.80

[

-32.00

10-3m]

[

-7.00

(c)

-8.67

2.20

(d)

-0.21

-10.33

-2.63

-12.00

-5.04

-13.67

-7.45

-15.33

-9.87

-17.00

-12.28

-18.67

-14.69

-20.33

-17.11

-22.00

-19.52

-23.67

-21.93

-25.33

-24.35

-27.00

V/Bcu = 2.77, H/Bcu = 1.00 θ = +20ƒ, Eu/cu=300 α=1.5, β=1.5, m=1, n=1

-26.76

V/Bcu = 2.75, H/Bcu = 1.00 θ = +20ƒ, Eu/cu=300 λ=1.0, m=1, n=1

-28.67 -30.33

-29.17 -31.59

-32.00

-34.00

[

[m]

-0.53

(f)

-57.70

-1.06

-121.82

-1.60

-185.94

-2.13

-250.06

-2.67

-314.18

-3.20

-378.30

-3.73

-442.42

-4.27

-506.54

-4.80

-570.66 -634.78

-5.33 -5.87

V/Bcu = 1.51, H/Bcu = 0.55 θ = +20ƒ, Eu/cu=300 α=1.5, β=0, m=1, n=2

-6.40 -6.93 -7.47

V/Bcu = 1.88, H/Bcu = 0.68 θ = +20ƒ, Eu/cu=300 α=1.5, m=1, n=1 Parallel, s=1.5

-8.00

[ 10-3m] 5.89

(g)

-11.48 -28.84 -46.21 -63.58 -80.94 -98.31 -115.68 -133.04 -150.41 -167.78

V/Bcu = 2.74, H/Bcu = 1.00 θ = +20ƒ, Eu/cu=300 α=1.5, m=1, n=1 Symmetrical, s=2

10-3m] 6.42

0.00

(e)

10-3m]

-185.15 -202.51 -219.88 -237.25 -254.61

Fig. 10. FE displacement contours for footing-void system with different geometric parameters.

-698.90 -763.02 -827.15 -891.27 -955.39

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Fig. 11. H–V failure envelops for spread footing above single rectangular void with constant height.

on the horizontal capacity of the footing. The inclination angles given by the expressions from Meyerhof [16] and Vesic [17] were 16.1° and 17.7°, respectively. 6. Results and discussion Numerical solutions for a range of values of the soil properties and the geometrical parameters were obtained from finite element (FE) analyses. The results are presented as failure envelopes in terms of the bearing capacity factors (H/Bcu and V/Bcu) and the horizontal and vertical loads normalized by the respective ultimate loads (H/Hult and V/Vult), referred to as the normalized loads. The former indicate the absolute size of the failure envelope, and the latter indicate the shape and the relative size of the failure envelope. Fig. 5 shows the effect of soil stiffness on the failure envelope for a footing above a single square void. The horizontal and vertical bearing capacities of the footing-above-void system are essentially independent of the soil stiffness. Previous research has shown that the soil stiffness has a negligible influence on the bearing capacity of a surface footing on level ground [24], which is consistent with the results shown in Fig. 5. Fig. 5(a) shows the failure envelopes plotted in terms of the bearing capacity factors. For soil without voids, the horizontal bearing capacity factor increases and the vertical bearing capacity factor decreases when the load inclination increases. The changes in both bearing capacity factors are very small for inclination angles greater than 20°. This behaviour implies that the horizontal load-carrying capacity is insensitive to the vertical load when the vertical load is light. The presence of a void reduces the failure envelope, with more significant reduc-

Fig. 12. H–V failure envelops for spread footing above single rectangular void with constant area.

tions in the failure loads occurring in conjunction with lower load inclinations. The critical angle of inclination is approximately 30° for a footing with a single square void with the geometric parameters a = 1.5 and b = 0. When expressed in terms of the normalized loads, Fig. 5(b) shows that the failure envelopes for footings in presence and absence of voids have distinctive shapes. However, the failure envelopes for footings without voids lie inside of those for footings with voids. In addition, it was observed that for footings over soil with and without voids, the horizontal load governs collapse for vertical loads less than approximately 0.5 Vult and 0.9 Vult, respectively, indicating that pure sliding occurs in these regions where H = Hult. Fig. 6 shows the failure envelopes for footings centred above a single square void for various values of a. Fig. 6(a) shows that the failure envelope expands with increasing a up to a limit (i.e., acr = 3.5), which represents the bearing capacity factor for a surface footing without voids. This behaviour indicates that the effect of the void location on the bearing capacity decreases as the distance between the footing and the void increases, and the effect vanishes when a reaches a value of 3.5. In fact, the expansion of the failure envelope with increasing a is caused by the increase in the vertical ultimate load, and the horizontal ultimate load does not contribute to the expansion of the failure loci. It was found that the failure envelopes are symmetric about the vertical load line at H/Bcu = 0, indicating that the footing bearing capacity is influenced by the magnitude of the load inclination, not by the sign (direction) of the load inclination (i.e., positive or negative values). As shown in Fig. 6(b), the shape of the failure envelope is similar but not unique for a < 3.5: although the actual load capacities are lower

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for a < 3.5, higher normalized loads are obtained and the largest failure envelope is achieved at a = 2.0. Fig. 7 shows the failure envelopes for footings above single square voids for various values of b and a constant value of a. As shown in Fig. 7(a), the expansion of the failure envelope demonstrates the increased bearing capacity with increasing b up to a limit (i.e., bcr = 2.5). Unlike the results in Fig. 6, it is observed that the failure envelopes are obviously asymmetric for positive and negative load inclinations. In general, the envelopes for positive load inclinations are larger than those for negative load inclinations, with the most significant difference in bearing capacity factors occurring at b = 0.5. This result can be explained by the two distinct kinematic mechanisms evident in the FE displacement contours (Fig. 8) that accompany failure with a load at an angle of h = ± 20° for b = 0.5. For a positive load inclination (h = + 20°), the collapse involves a combination of ceiling and wall failures, i.e., the collapse of the soil both above the void and adjacent to the wall of the void, whereas for a negative load inclination (h = 20°), collapse involves only a ceiling failure, i.e., the collapse of a portion of the soil immediately above the void. This indicates that the soil surrounding the void provides less support with a negative load inclination compared with a positive load inclination. Fig. 7(b) shows the failure envelopes normalized by the respective ultimate loads in the H–V plane. The envelopes expand and change shape considerably as the value of b varies. Fig. 9 shows the failure envelopes for footings above a single square void for various values of k. Fig. 9(a) indicates that the envelopes for footings with radial voids are in between those for footings with horizontal and vertical voids. As shown in Fig. 9(b), the normalized load curve for footings without voids always pro-

Fig. 14. H–V failure envelops for spread footing above double voids (symmetrical configuration).

Fig. 13. H–V failure envelops for spread footing above double voids (parallel configuration).

vides a safe estimate for the innermost failure envelope, which is consistent with the results in Figs. 6(b) and 7(b). Fig. 10(a–d) compare the FE displacement contours for single square voids at various locations. As the distance between the footing and void increases, it is likely that the region of collapse becomes wider and extends deeper. Fig. 11 shows the failure envelopes for rectangular voids for various values of the void width parameter n > 1 and a constant void height parameter (m = 1). Fig. 11(a) indicates that for a given value of n, higher values of a lead to higher vertical and horizontal bearing capacity factors, which are dictated by the expansion of the failure envelopes. It was also found that at a constant value of a, the overall size of the failure envelope gradually decreases as the value of n increases. Represented in the normalized load phase (Fig. 11(b)), the envelopes for a = 1.5 fall in a wide band, and the shapes of the envelopes differ at the shoulders as the value of n increases. However, the envelopes for a = 2.5 are relatively independent of the value of n. In addition, the FE displacement contours for the rectangular voids indicate that the failure mechanism is more extensive than a simple vertical translation of soil and involves greater soil rotation, as shown in Fig. 10(e). Fig. 12 shows the failure envelopes for rectangular voids for various combinations of m and n with a constant cross-sectional area, i.e., mn = 1. Fig. 12(a) indicates that as the void becomes narrower, the vertical and horizontal failure loads decrease for a given value of a. The normalized failure envelopes shown in Fig. 12(b) are similar to those in Fig. 11(b), which do not follow a unique curve. The failure envelopes for footings above double square voids with parallel and symmetrical configurations are compared in Figs 13 and 14 in terms of the bearing capacity factor and the normal-

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ized load, respectively. It was found that the envelopes are asymmetric for parallel configurations and symmetric for symmetrical configurations. As shown in Figs. 13(a) and 14(a), substantial variations that depend on the spacing between the two voids are evident: the bearing capacity of the footing increases as the distance between the two voids increases, regardless of the void configuration. This behaviour is attributed to the higher shearing resistance provided by the wider pillar between the adjacent voids. Fig. 13(a) shows that for two values of the vertical void distance parameter a = 1.5 and a = 2.5, there exists a critical spacing, scr = 1.5 and scr = 2, respectively, for the parallel configuration, indicating a spacing beyond which the interference effect of the voids becomes negligible. Fig. 14(a) shows that the critical spacing for symmetrical voids located at a = 1.5 is scr = 6. Additionally, the innermost envelope corresponds to that of a single rectangular void with parameters n = 2 and m = 1 (see Fig. 11). The normalized load diagrams are shown in Figs. 13(b) and 14(b). It is apparent that the shapes of the failure envelopes are affected by the spacing of the voids. Fig. 10(f) and (g) compare the failure patterns of footings overlying double square voids with parallel and symmetrical configurations. These results indicate that the footing has a greater bearing capacity for symmetrical void configurations than for parallel configurations. 7. Conclusions Small-strain finite element analyses of surface spread footings on undrained homogeneous clay with voids were performed to determine the undrained failure envelopes for the footings. The results of the FE analysis were validated by comparing them with other available solutions for the undrained bearing capacity of footings without voids. The effects of location, geometry and number (i.e., a single void and two voids located side by side) in addition to the soil stiffness were investigated. The following conclusions can be drawn from the present study: (1) The failure envelopes give a convenient and clear way to examine the undrained bearing capacity of a footing and provide a safety margin under any specific combination of horizontal and vertical loading. (2) The finite element analysis for the case of no voids in the soil compares well with Green’s closed-form solution. The soil stiffness has little effect on the combined bearing capacity of footings on clay soil with and without voids, indicating that the failure envelopes in the H–V plane follow a unique curve. (3) For single square voids, the absolute size of the failure envelope expands as the distance of the void from the footing in the vertical, horizontal and radial directions (a, b and k) increases. There exist certain critical void distances (acr = 3.5, bcr = 2.5 and kcr = 2.0) beyond which the failure envelopes do not change. The normalized failure envelopes (i.e., H/Hult and V/Vult) for footings with voids lie entirely inside those for footings without voids. For footings above voids, the failure envelopes take the form of symmetric ellipses that are functions of the location parameter a. In contrast, for footings above voids the failure envelopes are asymmetric functions of the location parameters b and k and depend on the direction of the load inclination.

(4) For single rectangular voids, the absolute size of the failure envelope contracts as the void becomes wider for a constant value of a. The shape and the relative size of the failure envelope are highly dependent on the geometry of the void. (5) For two square voids, the absolute size of the failure envelope increases with the spacing between the two voids. To eliminate the interference effect of the voids, the spacing between the two voids must be greater than a certain critical spacing. The critical spacing for parallel configurations is generally lower than that for symmetrical configurations.

Acknowledgements The authors would like to acknowledge the support of the National Research Foundation of Korea (NRF) (Grant Nos. 20110030040 and 2013R1A6A3A01023199) for this research. References [1] Truong QH, Eom YH, Lee JS. Stiffness characteristics of soluble mixtures. Geotechnique 2010;4:293–7. [2] Augarde CE, Lyamin AV, Sloan SW. Prediction of undrained sinkhole collapse. J Geotech Geoneviron Eng 2003;129:197–205. [3] Baus RL, Wang MC. Bearing capacity of strip footings above void. J Geotech Eng 1983;109:1–14. [4] Wood LA, Larnach WJ. The behaviour of footings located above voids. In: Proc 11th int conf on soil mech found eng; 1985. p. 273–6. [5] Al-Tabbaa A, Russell L, O’Reilly M. Model tests of footings above shallow cavities. Ground Eng 1989;22:39–42. [6] Sreng S, Ueno K, Mochizuki A. Bearing capacity of ground having a void. In: 57th JSCE annual meeting; 2002. p. 1221–2 [in Japanese]. [7] Badie A, Wang MC. Stability of spread footing above void in clay. J Geotech Eng 1984;110:1591–605. [8] Wang MC, Badie A. Effect of underground void on foundation stability. J Geotech Eng 1985;111:1008–19. [9] Azam G, Jao M, Wang MC. Cavern effect on stability of strip footing in twolayer soils. J Geotech Geoneviron Eng 1997;28:151–64. [10] Kiyosumi M, Kusakabe O, Ohuchi M, Peng FL. Yielding pressure of spread footing above multiple voids. J Geotech Geoneviron Eng 2007;133:1522–31. [11] Lee JK, Jeong S, Ko J. Undrained stability of surface strip footings above voids. Compos Geotech 2014;62:128–35. [12] Wang MC, Hsieh CW. Collapse load of strip footing above circular void. J Geotech Eng 1987;113:511–5. [13] Kiyosumi M, Kusakabe O, Ohuchi M. Model tests and analyses of bearing capacity of strip footing on stiff ground with voids. J Geotech Geoneviron Eng 2011;137:363–75. [14] Prandtl L. Uber die Harte Plasticher Korper. Gottingen Math Phys Kl 1920;12:74–85. [15] Hansen JB. A general formula for bearing capacity. Dan Geotech Inst 1961;11:38–46. [16] Meyerhof GG. Some recent research on the bearing capacity of foundations. Can Geotech J 1963;1:16–26. [17] Vesic AS. Analysis of ultimate loads of shallow foundations. J Soil Mech Found 1973;99:45–73. [18] Green AP. The plastic yielding of metal junctions due to combined shear and pressure. J Mech Phys Solids 1954;2:197–211. [19] Murff JD. Limit analysis of multi-footing foundation systems. In: Siriwardane HJ, Zaman MM, editors. Proc comp method adv geomech. Morgantown; 1994. p. 233–44. [20] Abbo AJ, Wilson DW, Sloan SW, Lyamin AV. Undrained stability of wide rectangular tunnels. Comp Geotech 2013;53:46–59. [21] Brinkgreve RBJ, Engin E, Swolfs WM. Plaxis user’s manual. Netherlands: Plaxis, BV; 2012. [22] Shiau JS, Merifield RS, Lyanmin AV, Sloan SW. Undrained stability of footings on slopes. Int J Geomech 2011;11:381–90. [23] Monajemi H, Razak HA. Finite element modelling of suction anchors under combined loading. Mar Struct 2009;22:660–9. [24] Chen Z, Tho KK, Leung CF, Chow YK. Influence of overburden pressure and soil rigidity on uplift behaviour of square plate anchor in uniform clay. Comput Geotech 2013;52:71–81.